A Normal Approximation for Binomial,F, Beta, and other Common, Related Tail Probabilities, II

Abstract This paper describes and derives the asymptotic behavior of the new Normal approximation introduced in Part I and of a family of Normal approximations based on roots, including the square root approximations of Fisher, and Freeman and Tukey, and the cube root approximations of Wilson and Hilferty, Camp, and Paulson. Various asymptotic comparisons are made, all of which rank the new approximation first, the cube root approximations second, and the other root approximations (and the ordinary Normal approximation) third. For instance, in the binomial case, if the tail probability is fixed as n → ∞, the errors resulting from the foregoing approximations are generally of order n -3/2, n -1, and n -1/2 respectively, while for a tail probability approaching 0, the relative error in approximating it approaches 0 if the corresponding standard Normal deviate is of smaller order than n 1/2 n 1/4, or n 1/6 respectively, but not generally otherwise. These comparisons are deduced from far more detailed results which are also given.